Chemical Chaos and Attractor Reconstruction

In this section we describe some beautiful experiments on the Belousov-Zhabotin-sky chemical reaction. The results show that strange attractors really do occur in nature, not just in mathematics. For more about chemical chaos, see Argoul et al. (1987). 

In the BZ reaction, malonic acid is oxidized in an acidic medium by bromate ions, with or without a catalyst (usually cerous or ferrous ions). It has been known since the 1950s that this reaction can exhibit limit-cycle oscillations, as discussed in Section 8,3. By the 1970s, it became natural to inquire whether the BZ reaction could also become chaotic under appropriate conditions. Chemical chaos was first reported by Schmitz, Graziani, and Hudson (1977), but their results left room for skepticism—some chemists suspected that the observed complex dynamics might be due instead to uncontrolled fluctuations in experimental control parameters. What was needed was some demonstration that the dynamics obeyed the newly emerging laws of chaos. 

The elegant work of Roux, Simoyi, Wolf, and Swinney established the reality of chemical chaos (Simoyi et al. 1982, Roux et al, 1983). They conducted an experiment on the BZ reaction in a continuous flow stirred tank reactor. In this standard set-up, fresh chemicals are pumped through the reactor at a constant rate to replenish the reactants and to keep the system far from equilibrium. The flow rate acts as a control parameter. The reaction is also stirred continuously to mix the chemicals. This enforces spatial homogeneity, thereby reducing the effective number of degrees of freedom. The behavior of the reaction is monitored by measuring S( ), the concentration of bromide ions. 

Roux et al. (1983) exploited a surprising data-analysis technique, now known as attractor reconstruction (Packard et al. 1980, Takens 1981), The claim is that for systems governed by an attractor, the dynamics in the full phase space can be reconstructed from measurements of just a single time series Somehow that single variable carries sufficient information about all the others. 

Roux et al. then constructed an approximate one-dimensional map that governs the dynamics on the attractor. Let Xj. X,, . . . denote successive values of B t -I- T) at points where the 

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